Theorem sbcieg 3782

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2175, ax-12 2212. (Revised by GG, 12-Oct-2024.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		df-sbc 3744	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		sbcieg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elabg 3634	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2142  {cab 2740  [wsbc 3743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3744

This theorem is used by:  sbcie  3784  2nreu  4408  reuprg0  4667  rabsnif  4688  ralrnmptw  7089  ralrnmpt  7091  fpwwe2lem3  10624  nn1suc  12261  opfi1uzind  14555  mndind  18893  fgcl  24046  cfinfil  24061  csdfil  24062  supfil  24063  fin1aufil  24100  ifeqeqx  32899  nn0min  33176  bnj1452  35449  cdlemk35s  41739  cdlemk39s  41741  cdlemk42  41743  2nn0ind  43700  zindbi  43701  prproropreud  48286